How do you find a tangent line parallel to secant line?

1 Answer
Oct 5, 2014

You can find a tangent line parallel to a secant line using the Mean Value Theorem.

The Mean Value Theorem states that if you have a continuous and differentiable function, then

f'(x) = (f(b) - f(a))/(b - a)f'(x)=f(b)f(a)ba

To use this formula, you need a function f(x)f(x). I'll use f(x) = -x^3f(x)=x3 as an example.

I'll also use a = -2a=2 and b = 2b=2 for the interval for the secant line. This is the line that passes through the points (-2, 8)(2,8) and (2, -8)(2,8).

So, we know that the slope of this line will be (-8 - 8)/(2 - (-2)) = -4882(2)=4.

To find the tangent lines parallel to this secant line, we will take the function's derivative, f'(x)f'(x), and set it equal to -44, then solve for xx.

-3x^2 = -43x2=4

Solving this for xx gives us: x = ±sqrt(4/3)x=±43.

So, the lines tangent to y = -x^3y=x3 at x = sqrt(4/3)x=43 and x = -sqrt(4/3)x=43 must be parallel to the secant line passing through x = 2x=2 and x = -2x=2.